Special relativity, no less
than Galilean relativity, presupposes a three-dimensional space, within which
individual physical entities exist and move over time. Each localized entity
(such as a particle or a point of a field) has, at any given time, a position
in space that can be characterized by three real numbers, representing the
spatial coordinates of the entity, and all the spatial relations between
entities can be inferred from these coordinates. Hence, given N point-like
entities, the N(N-1)/2 spatial distances between pairs of entities (at
a given time) can be encoded as the 3N coordinates of those particles. From a
purely relational point of view, without presupposing any embedding in a
manifold with a fixed number of dimensions, we have no reason to expect such
a reduction in the degrees of freedom. Each of the N(N-1)/2 pair-wise separations
could be regarded as an independent quantity. The apparent fact that these
separations are not independent, but can be encoded in the form of N sets of
three numbers, which vary continuously with time, provides the justification
for the idea of particles moving in a coherent three-dimensional space, and is
one of the strongest arguments against the possibility of a purely relational
basis for physical theories – unless we include space (or spacetime) itself
as a dynamical entity.

If we insist on a purely
relational theory, with no background space, one could argue that the separations
between entities ought to be regarded as the primary ontological entities, with
the particles serving merely abstract concepts to organize our knowledge of those
separations. The relationist view doesn’t even presuppose a definite
dimensionality of space, since each “separation” could be considered to
represent an independent degree of freedom, absent any additional restricting
principles. Of course, this freedom doesn’t seem to exist in the real world,
since (for example) we cannot arrange five particles all mutually equidistant
from each other. Nevertheless, it’s an interesting exercise to focus on the
spatial separations that exist between material particles, rather than on the
space and time coordinates of individual particles, to see if the behavior of
these separations can be characterized in a simple way.

From the conventional point
of view, the simplest motion is that of a free particle moving inertially,
i.e., in a straight line at uniform speed, which can be described by saying
that the space coordinates are linear functions of the time coordinate
(assuming a system of inertial coordinates). From a relational standpoint, we
consider the spatial separation between two such particles. The three
orthogonal components Dx, Dy, and Dz of the separation are linear functions of time, i.e.,

where the coefficients ai
and bi are constants. Therefore the magnitude of any such "co-inertial”
separation is of the form

where

Letting the subscript n
denote nth derivative with respect to time, the first two derivatives of s(t)
are

The right hand equation
shows that s2 s03 = k, and we can
differentiate this again and divide the result by s02
to show that the separation s(t) between any two particles in relatively
unaccelerated (i.e., co-inertial) motion in Galilean spacetime must satisfy
the equation

Next, consider a particle
of mass m attached to a rod in such a way that it can slide freely along the
rod. If we rotate the rod about some fixed point, the particle will tend to
slide outward along the rod away from the center of rotation in accord with
the basic equation of motion

where s is the distance
from the center of rotation to the sliding particle, and w is the angular velocity of the rod. Differentiating and multiplying
through by s0 gives

Then since s2 = w2s0,
we see that s(t) satisfies the equation

which is formally similar
to (1).

For a final example,
consider the separation between two massive particles in gravitational
free-fall due to their mutual gravitational attraction. Assume the two
particles are identical, each of mass m, lying along a line that is not
rotating. According to Newtonian theory the equation for this separation is

where G is a universal
constant. Note that each particle's "absolute" acceleration is half
of the second derivative of their mutual separation with respect to time. Re-arranging
terms, we have s2 s02 = -2Gm. Differentiating this again and dividing through by s0,
we can characterize radial gravitational free-fall by the purely kinematic
equation

So, we find once again that
a common (albeit idealized) class of physical separations satisfies the same
form of differential equation, even though in this case the separation is
governed by gravitation rather than just kinematics and inertia. All these
separations are characterized by an equation of the form

for some constant N. (Among
the other solutions of this equation, with N = -1, are the
elementary transcendental functions et, sin(t), and cos(t).
Solving for N, to isolate the arbitrary constant, we have

Differentiating both sides,
we get the basic equation

If none of s0, s1,
s2, and s3 is zero, we can divide each term by all of
these to give the aesthetically appealing form

This could be seen as a
(admittedly very simplistic) “unification” of a variety of physically
meaningful spatial separations under a single equation. However, this
obviously doesn’t encompass more than a tiny fraction of the variety of
spatial separations in the physical world. In fact, it’s clear that the
objects of our experience, viewed in isolation, cannot possibly be fully
characterized by just their mutual separations. For example, the separation
between two massive particles, as discussed above, beginning from a given
stationary value, will shrink to zero due to gravitational collapse, but only
if the system of particles has no angular momentum. If the particles are
revolving about their common center of mass at a suitable speed, the separation
between them can remain constant. Thus we have distinct outcomes for a single
intrinsic configuration. This is similar to Newton’s thought experiment
with the rotating pail of water, and it is another reason that Newton
founded his physics on the idea of absolute space, rather than on a purely
relational basis. Of course, as Mach observed, the evident physical effects
of “absolute rotation” don’t necessarily refute relationism as a viable basis
for coordinating events. It may be that we must take more relations into
account. For example, even though the two revolving particles are
intrinsically identical to two stationary particles, the configurations are
distinct when the relations of the particles to surrounding objects are
considered. From this point of view, we wouldn’t expect to be able to treat
individual separations, or even a limited cluster of related separations, in
isolation. It would presumably be necessary to account for all separations in
the universe in order to correctly analyze any part of the universe. This
might seem to make relationism hopeless, but most attempts to construct such
a theory have invoked simplifying assumptions about the aggregate effects of
very distant separations.

Newton’s achievement was finding a way to analyze isolated
parts of the world without having to explicitly refer to the rest of the
universe. All the effects of the distant universe can be encoded in the
simple concept of inertia in absolute space and time. To illustrate this
approach, and to show how it relates to the separation equation discussed
above, consider the general Newtonian equation of motion of a particle in a
stationary spherical gravitational field:

In these equations, r is
the magnitude of the distance of the particle from the center of the field
and w is the absolute angular velocity of the
particle. A single coherent system of coordinates, with a single definition
of absolute rotation, suffices for the analysis of all physical systems. We
have no right to expect this to be true, but our experience has taught us
that it is true. Now, if we solve the left hand equation for w and differentiate to give dw/dt, we can
substitute these expressions into the right hand equation and re-arrange the
terms to give

Incidentally, even though
the above has been based on the Galilean spatial separations between objects
as a function of Galilean time, the same conditions can be shown to apply to
the absolute spacetime intervals between inertial particles as a function of
their proper times. Relative to any point on the worldline of one particle,
the four components Dt, Dx, Dy, and Dz of the absolute interval to any other inertially
moving particle are all linear functions of the proper time t along the latter particle's worldline. Therefore, the components can
be written in the form

where the coefficients ai
and bi are constants. It follows that the absolute magnitude of
any "co-inertial separation" is of the form

where

Thus we have the same formal
dependence as before, except now the parameter s represents the absolute
spacetime separation. This shows that the absolute separation between any
fixed point on one inertial worldline and a point advancing along any other
inertial worldline satisfies equation (1), where subscripts denote
derivatives with respect to proper time of the advancing point. Naturally the
reciprocal relation also holds, as well as the absolute separation between
two points, each advancing along arbitrary inertial worldlines, correlated
according to their respective proper times.